Problem: All of the 3rd grade teachers and students from Loyola went on a field trip to an art museum. Tickets were $$7.00$ each for teachers and $$2.50$ each for students, and the group paid $$29.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$21.00$ each for teachers and $$11.50$ each for students, and the group paid $$111.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+2.5y = 29}$ ${21x+11.5y = 111}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-21x-7.5y = -87}$ ${21x+11.5y = 111}$ Add the top and bottom equations together. $ 4y = 24 $ $ y = \dfrac{24}{4}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {7x+2.5y = 29}$ to find $x$ ${7x + 2.5}{(6)}{= 29}$ $7x+15 = 29$ $7x = 14$ $x = \dfrac{14}{7}$ ${x = 2}$ You can also plug ${y = 6}$ into $ {21x+11.5y = 111}$ and get the same answer for $x$ ${21x + 11.5}{(6)}{= 111}$ ${x = 2}$ There were $2$ teachers and $6$ students on the field trips.